Exact Asymptotic Results for Persistence in the Sinai Model with Arbitrary Drift

TL;DR

This paper derives exact asymptotic results for the persistence of a biased Brownian particle in a Sinai landscape, revealing complex behaviors across different drift values using a novel quantum Hamiltonian mapping.

Contribution

Introduces a new method that maps persistence calculations to quantum Hamiltonian spectra, enabling analysis for arbitrary drift in Sinai models.

Findings

Persistence exhibits rich asymptotic behaviors with qualitative changes at specific drift values.

Method allows analytical results beyond the zero-drift limit.

Reveals complex drift-dependent persistence phenomena.

Abstract

We obtain exact asymptotic results for the disorder averaged persistence of a Brownian particle moving in a biased Sinai landscape. We employ a new method that maps the problem of computing the persistence to the problem of finding the energy spectrum of a single particle quantum Hamiltonian, which can be subsequently found. Our method allows us analytical access to arbitrary values of the drift (bias), thus going beyond the previous methods which provide results only in the limit of vanishing drift. We show that on varying the drift, the persistence displays a variety of rich asymptotic behaviors including, in particular, interesting qualitative changes at some special values of the drift.